Question

If \(K_{c}=0.042\) for \(\mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{PCl}_{5}(g)\) at \(500 \mathrm{~K}\), what is the value of \(K_{p}\) for this reaction at this temperature?

Short answer

The value of \(K_p\) for the reaction \(PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g)\) at 500 K is approximately 0.001023.

Step-by-step solution

Write down the balanced chemical equation

The balanced equation for the given reaction is: \(PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g)\)

Determine Δn for the reaction

To find Δn, we simply subtract the number of moles of gaseous reactants from the number of moles of gaseous products: Δn = (moles of PCl5) - (moles of PCl3 + moles of Cl2) Δn = 1 - (1 + 1) = -1

Calculate Kp

Now, we will use the relationship between Kc and Kp, along with the given Kc value and the calculated Δn value, to find the value of Kp: \(K_p = K_c(RT)^{(\Delta n)}\) where, Kc = 0.042 R = 0.08206 L atm / mol K T = 500 K Δn = -1 \(K_p = 0.042 \times (0.08206 \times 500)^{-1}\) \(K_p = 0.042 \times (41.03)^{-1}\) \(K_p = 0.042 \times 0.02437\) \(K_p = 0.001023\) So, the value of Kp for this reaction at 500 K is approximately 0.001023.

Key concepts

Chemical Equilibrium

Understanding the concept of chemical equilibrium is a foundational step in mastering the behavior of reactions. In a physical sense, when a reaction reaches a state of chemical equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentration of reactants and products. At this point, the reaction can appear to be inactive, but in reality, the chemical processes are dynamically balanced.

For the equation given in our exercise - \(PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g)\), the equilibrium is established between the reactants - phosphorus trichloride and chlorine gas, and the product - phosphorus pentachloride. To quantify the position of equilibrium, we use an equilibrium constant, in this case, \(K_{c}\), which is calculated using the concentrations of the reactants and products raised to the power of their coefficients in the balanced equation. A high value of \(K_{c}\) implies a greater concentration of products at equilibrium, favoring the forward reaction, while a low value suggests a dominance of reactants.

Relation Between Kc and Kp

When dealing with gas-phase reactions, it's essential to understand the relationship between the equilibrium constants \(K_{c}\) and \(K_{p}\). \(K_{c}\) is related to the concentration or molarity of the reactants and products, while \(K_{p}\) is related to the partial pressures. The two are connected through the equation:
\[K_{p} = K_{c}(RT)^{\Delta n}\]
where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in the number of moles of gas in the reaction. If \(\Delta n\) is positive, it means there are more moles of gaseous products than reactants, which results in the value of \(K_{p}\) being greater than \(K_{c}\). Conversely, if \(\Delta n\) is negative, there are fewer moles of gaseous products, and \(K_{p}\) will be less than \(K_{c}\), as exhibited in the exercise where \(\Delta n\) was -1. This equation provides a bridge between the molarity-based and pressure-based descriptions of chemical equilibrium for gases.

Gas-Phase Reaction

Gas-phase reactions present specific considerations that differ from reactions in other states, such as those in solution. The properties of gases, their behavior under varying temperatures and pressure, and the way they occupy volume are described by the ideal gas law. Particularly in the context of equilibrium, we focus on partial pressures, which are directly proportional to the molar concentrations at a constant temperature according to the ideal gas law (PV = nRT).

When a gas-phase reaction reaches equilibrium, as with the synthesis of \(PCl_{5}(g)\) from \(PCl_{3}(g)\) and \(Cl_{2}(g)\), the equilibrium constant expressed in terms of partial pressure (\(K_{p}\)) becomes particularly relevant. This is because changes in pressure can shift the equilibrium position, as Le Chatelier's principle tells us that a system at equilibrium will adjust to minimize the effects of any changes imposed upon it. Therefore, for gas-phase reactions, understanding how to use and calculate \(K_{p}\) is critical for predicting the behavior of the reaction under various conditions.